A set \(K\subseteq \mathbb {T}\)  is a Kronecker set if for every continuous function \(f:K\rightarrow \mathcal {S}^{1}\)  and all \(\varepsilon >0\) , there is an exponential function \(e_{n}(x)=\exp (2\pi i n x)\)  such that \(\sup _{x\in K}|f(x)-e_{n}(x)| < \varepsilon \) . The group \(\mathbb {T}\)  contains a nonempty compact Kronecker set with no isolated points ([1]}; see also [2]}, [3]}). In other words, \(\mathbb {T}\)  contains a nonempty perfect Kronecker set. Consequently, there are continuous probability measures supported on Kronecker sets.
